About this video
To describe curvature mathematically, describe the curve parametrically with a vector-valued function s(t), such as (t − sin t, 1 − cos t). Then look at the unit tangent vector T at each point and ask how quickly it changes direction. Curvature is the magnitude of the rate of change of T with respect to arc length s, |dT/ds|, not with respect to the parameter t, so the answer does not depend on how fast you travel.
You'll be able to
- recall that sharp turns mean small radius of curvature and large curvature.
- describe a curve with a vector-valued function s(t).
- explain that curvature measures how quickly the unit tangent vector changes direction.
- explain why curvature uses the rate of change with respect to arc length and why we take its magnitude.
Key words
- parametric curve
- A curve traced by the tip of a vector s(t) as the parameter t varies.
- unit tangent vector
- A vector of length one that points along the curve at a given point, written T.
- arc length
- Distance measured along the curve itself, written s.
- curvature
- The magnitude of the rate of change of the unit tangent vector with respect to arc length, |dT/ds|.
Transcript
- [Voiceover] So, in the last video I talked about curvature and the radius of curvature, and I described it purely geometrically where I'm saying, you imagine driving along a certain road, your steering wheel locks, and you're wondering what the radius of the circle that you draw with your car, you know through whatever surrounding fields there are on the road as a result, and the special symbol that we have for this, for this idea of curvature is a little kappa, and that's gonna be one divided by the radius, and the reason we do that is basically you want a large kappa, a large curvature to correspond with a sharp turn so, sharp turn, small radius, large curvature.
But the question is how do we describe this in a more mathematical way? So, I'm gonna to go ahead and clear up, get rid of the circle itself and all of that radius and just be looking at the curve itself in it's own right. The way you typically describe a curve like this is para-metrically, so, you'll have some kind of vector valued function s that takes in a single parameter t, and then it's gonna output the x and y coordinates, each as functions of t.
So, this will be the x coordinate, the y coordinate. And then this specific case, I'll just tell you the curve that I drew happens to be parameterized by one minus the sign of t as the x component function, actually no, it's t minus sign of t, and the bottom part is one minus cosign of t.
That's the curve that I drew. And the way that you're thinking about this is for each value t you get a certain vector that puts you at a point on the curve, and as t changes the vector you get changes, but the tip of that vector kind of traces out the curve as a whole, right?
And you can imagine the vector drawing the curve as t varies, and the thought behind making curvature mathematical, here I'll kind of clear up some room for myself, is that you take the tangent vectors here so, so, you might image a unit tangent vector at every given point, and you're wondering how quickly those guys change direction, right?
So with the little schematic that I have drawn here I'm just gonna call this guy t one for like the first tangent vector, t2 , t3, and I haven't specified where they started or anything I just what give a feel for you've got various different tangent vectors and I'm just gonna say that all of them each one of them t sub somethings has a unit length.
And the idea of curvature is to look at how quickly that unit tangent vector changes directions. So, you know you might imagine a completely different space so, rather than rooting each vector on the curve, let's see what it would look like if you just kind of write each vector in its own right off in some other spot.
So, this guy here would be t one, and then t two points a little bit down, and then t three points kind of much more down So, all of these, this would be t one, that guy is t two and these are the same vectors I'm just kind of drawing them all rooted at the same spot so it's a little easier to see how they turn, and you wanna say, okay how much do you change as you go from t one to t two, is that a large angle change?
and as you go from t two to t three, is that a large change as well? And you can kind of see how if you have a curve, and let's say it's, if you have a curve that curves quite a bit, you know it's doing something like that then the unit vector, the unit tangent vector at this point changes quite rapidly over a short distance to be something almost 90 degrees different.
Whereas, if you take the unit vector here and see how much does it change as you go from this point over to this point it doesn't really change that much. So, the thought behind curvatures, we gonna take the rate of change of that unit tangent vector, so, the rate of change of t, and I'm gonna let capital T be a function that tells you whatever the unit tangent vector at each point is, and I'm not gonna take the rate of change in terms of, you know the parameter little t, which is what we use to parameterize the curve, because it shouldn't matter how you parameterize the curve, maybe you're driving along it really quickly or really slowly.
Instead, what you wanna take is the rate of change with respect to arc length. Arc length, and I'm using the variable s here to denote arc length, and what I mean by that is that if take just a tiny little step here the distance of that step, the actual distance in the x y plane, you consider to be the arc length.
And if you imagine it being really, really small you're considering that a d s, a tiny change in the arc length. So this is the quantity that we care about. How much does that unit tangent vector change with respect to a tiny change in the arc length? You know, as we travel along, let's say, it was a distance of like 0.5, right, you wanna know, did this unit vector change a lot or a little bit.
But I should add something here, this tiny change in the vector that would tell you, you know what the vector connecting their two tips is, so, this would be a vector valued quantity, and curvature itself should just be a number. So, what we really care about is the size of this guy.
So, what we really care about, the size of this, which is a vector valued quantity, and that'll be an indication of just how much the curve curves. But if I'm at a sharper turned curve you go over that same distance and then suddenly the change in the tangent vectors goes by quite a bit that would be telling you it's a high curvature.
And in the next video I'm gonna talk through what that looks like, how you think about this tangent vector function, this unit tangent vector function, and what it looks like to take the derivative of that with respect to arc length. It can get a little convoluted in terms of the symbols involved, and the constant picture you should have in the back of your mind is that circle, that circle that's hugging the curve very closely at a certain point, and this means of taking the magnitude of the rate of change of the unit tangent vector with respect to arc length is all just trying to capture that idea.
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